Method for determining the inflow profile of fluids of multilayer deposits

ABSTRACT

A method for determining the profile of fluids inflowing into multi-zone reservoirs provides for a temperature measurement in a wellbore during the return of the wellbore to thermal equilibrium after drilling and determining a temperature of the fluids inflowing into the wellbore from each pay zone after perforation at an initial stage of production. Specific flow rate for each pay zone is determined by a rate of change of the measured temperatures.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a U.S. National Stage Application under 35 U.S.C.§371 and claims priority to Patent Cooperation Treaty Application No.PCT/RU2012/000872 filed Oct. 25, 2012, which claims priority to RussianPatent Application No. RU2011143218 filed Oct. 26, 2011. Both of theseapplications are incorporated herein by reference in their entireties.

FIELD OF THE DISCLOSURE

The disclosure relates to the field of geophysical studies of oil andgas wells, in particular to determining the inflow profile of fluidsinflowing into the wellbore from multi-zone reservoirs.

BACKGROUND OF THE DISCLOSURE

Usually when estimating flow rate of individual pay zones by temperaturedata, temperature measurement along the entire wellbore is conducted,while temperature of a reservoir near the wellbore is assumed close tothe temperature of the undisturbed reservoir.

In particular, a method for determining relative flow rates of pay zonesby quasi-stationary flow temperatures measured along a wellbore isknown. This method is, for example, described in Cheremsky, G. A.Applied Geothermics, Nedra, 1977, p. 181. The main assumption of thetraditional approach is that an undisturbed temperature of a reservoirnear a wellbore is known prior to the tests. This assumption is notperformed if temperature is measured at a first stage of productionshortly after perforation of the well. The influence of the perforationitself is not very significant, but as a rule the temperature of thenear-wellbore part of formation is considerably lower than thetemperature of the undisturbed reservoir due to the cooling resultingfrom previous technological operations: drilling, circulation andcementing.

SUMMARY OF THE DISCLOSURE

The method for determining a profile of fluid inflow from a multi-zonereservoir provides the possibility to determine the inflow profile at aninitial stage of production, just after perforating a well, and inenhancing the accuracy of inflow profile determination due to thepossibility of determining inflow profile by transient temperature data.

The method comprises measuring temperature in a wellbore during awellbore-return-to-thermal-equilibrium time after drilling and thenperforating the wellbore. Temperature of fluids inflowing into thewellbore from pay zones is determined at an initial stage of productionand a specific flow rate for each pay zone is determined by rate ofchange of the measured temperatures.

In case of direct measurement of temperature of the fluids inflowinginto the wellbore from each pay zone, specific flow rate of each payzone is determined by the formula

${Q_{i} = {\frac{4\pi}{\chi} \cdot a \cdot h_{i} \cdot \left( {\frac{\overset{.}{T_{{in},i}}}{{\overset{.}{T}}_{s}} - 1} \right)}},$

where Q_(i) is a flow rate of an ith pay zone,

{dot over (T)}s is a rate of temperature recovery in the wellbore beforeperforation,

{dot over (T)}_(in), i is a rate of change of temperature of the fluidinflowing into the wellbore from the ith pay zone at an initial stage ofproduction,

h_(i) is a thickness of the ith pay zone,

a is a thermal diffusivity of a reservoir,

${\chi = \frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}},$

ρ_(f)c_(f) is a volumetric heat capacity of the fluid,

ρ_(r)c_(r)=φ·ρ_(f)c_(f)+(1−φ)·ρ_(m)c_(m) is a volumetric heat capacityof the rock saturated with the fluid,

ρ_(m)c_(m) is a volumetric heat capacity of a rock matrix;

φ is a porosity of the reservoir.

In situations where it is not possible to directly measure temperatureof the fluids inflowing into the wellbore from each pay zone,temperature of the fluids is determined with the use of sensorsinstalled on a tubing string, above each perforated interval. A specificflow rate of a lower zone is determined by the formula

${Q_{1} = {\frac{4\pi}{\chi} \cdot a \cdot h_{1} \cdot \left( {\frac{\overset{.}{T_{1}}}{{\overset{.}{T}}_{s}} - 1} \right)}},$

where Q₁ is a flow rate of a lower pay zone,

{dot over (T)}_(s) is a rate of temperature recovery in the wellborebefore perforation,

{dot over (T)}₁ is a rate of change of temperature of the fluidinflowing into the wellbore from the pay zone at an initial stage ofproduction as measured above the lower perforated interval,

h₁ is a thickness of this pay zone,

a is a thermal diffusivity of a reservoir,

${\chi = \frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}},$

ρ_(f)c_(f) is a volumetric heat capacity of the fluid,

ρ_(r)c_(r)=φ·ρ_(f)c_(f)+(1−φ)·ρ_(m)c_(m) is a volumetric heat capacityof the rock saturated with the fluid,

ρ_(m)c_(m) is a volumetric heat capacity of a rock matrix;

φ is a porosity of the reservoir.

Then with temperatures measured by the sensors installed on the tubingstring, specific flow rates of overlying zones are determined, usingvalues of flow rates determined for the underlying zones.

The wellbore return-to-thermal-equilibrium time usually lasts for 5-10days.

Temperature of the fluids inflowing into the wellbore from pay zones atthe initial state of production is measured within 3-5 hours from startof production.

BRIEF DESCRIPTION OF THE FIGURES

The disclosure is illustrated by drawings where:

FIG. 1 shows a scheme with three perforated intervals and threetemperature sensors;

FIGS. 2 a and 2 b show results of calculation of inflow profiles for twoversions of formation permeabilities;

FIG. 3 shows temperatures of fluids inflowing into the wellbore andtemperatures of the corresponding sensors for the case illustrated inFIG. 2 a;

FIG. 4 shows temperatures of the fluids inflowing into the wellbore andtemperatures of the corresponding sensors for the case illustrated inFIG. 2 b;

FIG. 5 shows time derivatives of fluid temperature and temperature ofsensor 1 for the case illustrated in FIG. 2 a;

FIG. 6 shows time derivatives of fluid temperature and temperature ofsensor 1 for the case illustrated in FIG. 2 b;

$f_{21} = {{\frac{{\overset{.}{T}}_{2}}{{\overset{.}{T}}_{1}}\mspace{14mu} {and}\mspace{14mu} f_{32}} = \frac{{\overset{.}{T}}_{3}}{{\overset{.}{T}}_{21}}}$

FIG. 7 shows ratios of temperature growth rates for FIG. 5;

FIG. 8 shows the same ratios for FIG. 6; and

FIG. 9 shows correlation between the time derivative T_(in) and specificflow rate q.

DETAILED DESCRIPTION

The method may be used with a tubing-conveyed perforation. It is basedon the fact that a near-wellbore space, as a result of drilling, usuallyhas a lower temperature than the temperature of surrounding rocks.

After drilling of a wellbore, circulation and cementing, temperature ofa reservoir in a near-borehole zone is (by 10-20 K and more) lower thanan original temperature of the surrounding reservoir at a depth underconsideration. After these stages, a relatively long period ofwellbore-returning-to-thermal-equilibrium follows during which otherworking operations in the well are carried out, including installationof a testing string with perforator guns. In the process ofwellbore-returning-to-thermal-equilibrium after drilling resultingcooling of near-wellbore formations, temperature measurements in thewellbore are conducted.

After perforation, an initial stage of production follows—cleanup of thenear-borehole zone of the reservoir. At the initial stage of production,when a change takes place in the temperature of fluids inflowing intothe wellbore (usually during 3-5 hours), temperature of the fluidsinflowing into the wellbore form each pay zone is measured.

In case of a homogeneous reservoir, radial profile of temperature in thereservoir prior to start of the cleanup is determined with the use ofsome general relationship that follows from the equation of conductiveheat transfer (1).

$\begin{matrix}{\frac{\partial T}{\partial t} = {a \cdot \left( {\frac{\partial^{2}T}{\partial r^{2}} + {\frac{1}{r} \cdot \frac{\partial T}{\partial r}}} \right)}} & (1)\end{matrix}$

where “a” is a heat diffusivity of the reservoir.

From the physical viewpoint, it will be justifiable to suppose that witha long wellbore-returning-to-thermal-equilibrium time, somenear-wellbore zone (r<r_(c)) exists within which the rate of increase oftemperature in the formation is constant, i.e. it does not depend ondistance from the wellbore:

$\begin{matrix}{\frac{\partial T}{\partial t} \approx {\phi (t)} \equiv {\overset{.}{T}}_{s}} & (2)\end{matrix}$

Equations (1) and (2) have the following boundary conditions at thewellbore axis:

$\begin{matrix}{{{T\left( {r = 0} \right)} = T_{a}};{\left. \frac{T}{r} \right|_{r = 0} = 0}} & (3)\end{matrix}$

where T_(a) is temperature at the axis (r=0).

The solution of the problem (1), (2), (3) is

T(r)≈T_(a)+b·r²   (4)

where

$\begin{matrix}{b = {\frac{1}{4 \cdot a} \cdot {\overset{.}{T}}_{s}}} & (5)\end{matrix}$

Formulas (4), (5) give an approximate radial temperature profile nearthe wellbore prior to start of production. A numerical simulationdemonstrates that after 50 hours ofborehole-return-to-thermal-equilibrium time, these formulas are adequatefor r<0.5-0.7 m (with accuracy of 1-5%) for an arbitrary possibleinitial (before closure) temperature profile.

Formulas (4), (5) do not take into consideration the influence of heatemission in the course of perforation and radial non-uniformity ofthermal properties of the wellbore and the reservoir, that is why aftercomparison with results of numerical simulation, introduction of somecorrection coefficient might be necessary.

After the start of production, the radial profile of the temperature inthe reservoir and transient temperatures of the produced fluid isdetermined, mainly, by convective heat transfer that is determined bythe formula

$\begin{matrix}{{{{\rho_{r}{c_{r} \cdot \frac{\partial T}{\partial t}}} - {\rho_{f}{c_{f} \cdot v \cdot \frac{\partial T}{\partial r}}}} = 0}{where}} & (6) \\{v = \frac{q}{2{\pi \cdot r}}} & (7)\end{matrix}$

is a velocity of radial filtration of the fluid, q [m³/m/s] is aspecific flow rate, ρ_(f)c_(f) is a volumetric heat capacity of thefluid, ρ_(r)c_(r)=φ·ρ_(f)c_(f)+(1−φ)·ρ_(m)c_(m) is a volumetric heatcapacity of the rock saturated with the fluid, ρ_(m)c_(m) is avolumetric heat capacity of the rock matrix, φ is a porosity of thereservoir.

Equation (6) does not account for conductive heat transfer, theJoule-Thomson effect and the adiabatic effect. The influence of theconductive heat transfer will be accounted for below, while theJoule-Thomson effect (ΔT=ε₀ΔP) and the adiabatic effect are small due toa small pressure differential ΔP and a relatively big typical cooling ofthe near-wellbore zone (5-10 K) before start of production.

Equation (6) has the following solution

$\begin{matrix}{{{T\left( {r,t} \right)} = {T_{0}\left( \sqrt{r^{2} + {\frac{\chi}{\pi}{q \cdot t}}} \right)}},} & (8)\end{matrix}$

where T₀(r) is an initial temperature profile in the reservoir (4),

$\chi = {\frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}.}$

Temperature of the fluid inflowing into the wellbore is (4), (8):

$\begin{matrix}{{{T_{in}(t)} = {T_{0}\left( \sqrt{r_{w}^{2} + {\frac{\chi}{\pi}{q \cdot t}}} \right)}}{or}{{{T_{in}(t)} \approx {T_{a} + {b \cdot \left( {r_{w}^{2} + {\frac{\chi}{\pi} \cdot q \cdot t}} \right)}}} = {\alpha + {\beta \cdot q \cdot t}}}{where}} & (9) \\{{\alpha = {T_{a} + {{\overset{.}{T}}_{s} \cdot \frac{r_{w}^{2}}{4{\pi \cdot a}}}}},} & (10) \\{\beta = {{\overset{.}{T}}_{s} \cdot {\frac{\chi}{4{\pi \cdot a}}.}}} & (11)\end{matrix}$

In accordance with (9), rate of fluid temperature increase at the inletis

$\frac{T_{in}}{t} = {\beta \cdot {q.}}$

This formula for rate of temperature increase of the produced fluid isnot fully correct because Equation (6) does not take into considerationthe conductive heat transfer. Even in cases of very small productionrates (q→0), temperature of the inflow increases due to the conductiveheat transfer and the approximate formula accounting for this effect canbe written in the following way

$\begin{matrix}{\frac{T_{in}}{t} = {{\beta \cdot q} + {\overset{.}{T}}_{s}}} & (12)\end{matrix}$

Thus, with direct measurement of temperature of the fluid inflowing intothe well, specific flow rate of each pay zone Q_(i) can be determined bythe formula

$\begin{matrix}{{Q_{i} = {\frac{4\pi}{\chi} \cdot a \cdot h_{1} \cdot \left( {\frac{{\overset{.}{T}}_{{in},i}}{{\overset{.}{T}}_{s}} - 1} \right)}},} & (13)\end{matrix}$

For cases where no possibility exists to directly measure temperature ofthe fluids inflowing into the wellbore from the pay zones, it issuggested to use results of temperature measurements above eachperforated interval, for example, with the use of sensors installed on atubing string utilized for perforating. In accordance with the numericalsimulation, in 20-30 minutes after start of production, the differencebetween temperature of the fluid inflowing into the wellbore T_(in,1)and temperature T₁ measured in the wellbore above a first perforatedinterval is practically constant: T_(in,1)−T=ΔT₁≈const , and

$\frac{T_{1}}{t} = {\frac{T_{{in},1}}{t}.}$

In accordance with Formula (12), this means that a flow rate of thelower pay zone Q₁ can be determined (Q₁=h₁·q₁) (h₁ is a thickness ofthis pay zone) by temperature measured above the first perforatedinterval:

$\begin{matrix}{{\overset{.}{T}}_{1} = {{\beta \cdot \frac{Q_{1}}{h_{1}}} + {\overset{.}{T}}_{s}}} & (14)\end{matrix}$

or, taking into consideration Formula (11), we find

$\begin{matrix}{Q_{1} = {\frac{4\pi}{\chi} \cdot a \cdot h_{1} \cdot \left( {\frac{{\overset{.}{T}}_{1}}{{\overset{.}{T}}_{s}} - 1} \right)}} & (15)\end{matrix}$

The parameters in this formula can be approximately estimated (“a” andχ) or measured. The value of {dot over (T)}_(s) is measured with the useof temperature sensors after installing the tubing string before theperforation. The value of {dot over (T)}₁ is measured above the firstperforation interval at the initial stage of production.

In case of three or more perforated zones, numerical simulation can beused for determining the inflow profile. For any set of values of flowrate {Q_(i)} (i=1, 2 . . . n, where n is the number of perforatedzones), transient temperatures of produced fluids can be calculated inthe following way (9):

$\begin{matrix}{T_{{in},i} = {\alpha_{i} + {\left( {{\beta \cdot \frac{Q_{i}}{h_{i}}} + {\overset{.}{T}}_{s}} \right) \cdot t}}} & (16) \\{{\alpha_{i} = {T_{a,i} + {{\overset{.}{T}}_{s} \cdot \frac{r_{w}^{2}}{4{\pi \cdot a}}}}},} & (17)\end{matrix}$

The parameter β (11) is one and the same for the zones; the parametersα_(i) are different because they depend on the temperature of thereservoir T_(a,i) recorded in the wellbore before start of production.

For this set of flow rate values, the numeric model of the producingwellbore should calculate transient temperatures of the flow at eachdepth of placement of the sensor with consideration of heat losses intothe surrounding reservoir, the calorimetric law for the fluids beingmixed in the wellbore, and the thermal influence of the wellbore whichis understood here as the influence of the fluid initially filling thewellbore. The flow rate is determined with the use of the procedure ofmodel fitting that minimizes differences between the recorded andcalculated temperatures of the sensors.

An approximate solution of the problem can be obtained with the use ofthe above-described analytical model, which utilizes rates of increaseof sensor temperatures.

The calorimetric law for the second perforated zone is described by theequation

$\begin{matrix}{\frac{{T_{1}^{*} \cdot Q_{1}} + {T_{{in},2} \cdot Q_{2}}}{Q_{1} + Q_{2}} = T_{2}^{*}} & (18)\end{matrix}$

where T₁* are T₂* are temperatures of the fluid below and above theperforated zone. In accordance with the numeric simulation, thedifference between T₁ and T₁*, T₂ and T₂* remains practically constantand instead of Equation (18) we can use the following equation for timederivatives of the measured temperatures:

$\begin{matrix}{\frac{{{\overset{.}{T}}_{1} \cdot Q_{1}} + {{\overset{.}{T}}_{{in},2} \cdot Q_{2}}}{Q_{1} + Q_{2}} = {\overset{.}{T}}_{2}} & (19)\end{matrix}$

Taking into consideration the above-presented relationships (11) and(16), this formula can be written as an equation for the dimensionlessflow rate y₂ of the second perforated zone y₂=Q₂/Q₁:

$\begin{matrix}{{{\frac{1}{1 + y_{2}}\left\{ {1 + {\left\lbrack \frac{{h_{12} \cdot y_{2}} + y_{a}}{1 + y_{a}} \right\rbrack \cdot y_{2}}} \right\}} = {\frac{{\overset{.}{T}}_{2}}{{\overset{.}{T}}_{1}} = f_{21}}}{{{{where}\mspace{14mu} h_{12}} = \frac{h_{1}}{h_{2}}},{y_{a} = {\frac{4{\pi \cdot a}}{\chi} \cdot {\frac{h_{1}}{Q_{1}}.}}}}} & (20)\end{matrix}$

If {dot over (T)}₂>{dot over (T)}{dot over (T₁)} (f₂₁>1), a uniquesolution exists. In the opposite version (f₂₁ <¹), this equation has twosolutions. The physical sense of this peculiarity is quite evident forf₂₁=1, that corresponding to equal increase rates of temperatures T₂ andT₁. Indeed, this may take place in two cases: (1) Q₂=0 (y₂=0) and abovethe upper zone the behavior of the temperature is the same as below it;(2) Q₂=Q₁ (y₂=1)—both zones are equal and they have the same rate oftemperature increase.

The possible solution of the problem of non-uniqueness of solution usesthe combination of two approaches. After evaluating Q₁ with the use ofEquation (12) and determining y₂ by Equation (20), the true value of y₂can be chosen using the known total flow rate Q (for two perforatedzones):

Q=Q ₁ +Q ₂ =Q ¹(1+y ₂)   (21)

Relative flow rates for perforated zones 3 and 4 can be calculated usingthe dimensionless values y₂, y₃ and so on, which were determinedpreviously for the perforated zones located down the wellbore.

$\begin{matrix}{{\frac{1}{1 + y_{3}}\left\{ {1 + {\left\lbrack \frac{{{y_{3}\left( {1 + y_{2}} \right)} \cdot h_{13}} + y_{a}}{\left( {1 + y_{a}} \right) \cdot f_{21}} \right\rbrack \cdot y_{3}}} \right\}} = f_{32}} & (22) \\{{{\frac{1}{1 + y_{4}}\left\{ {1 + {\left\lbrack \frac{{y_{4} \cdot \left\lbrack {1 + y_{2} + {y_{3}\left( {1 + y_{2}} \right)}} \right\rbrack \cdot h_{14}} + y_{a}}{\left( {1 + y_{a}} \right) \cdot f_{21} \cdot f_{32}} \right\rbrack \cdot y_{4}}} \right\}} = f_{43}}{where}{{y_{3} = \frac{Q_{3}}{Q_{1} + Q_{2}}},{y_{4} = \frac{Q_{4}}{Q_{1} + Q_{2} + Q_{3}}},{f_{32} = \frac{{\overset{.}{T}}_{3}}{{\overset{.}{T}}_{2}}},{f_{43} = {\frac{{\overset{.}{T}}_{4}}{{\overset{.}{T}}_{3}}.}}}} & (23)\end{matrix}$

The possibility of determining the inflow profile with the use of thesuggested method for a case where direct measurement of temperatures offluids inflowing into the wellbore from pay zones is impossible waschecked up on synthetic examples prepared with the use of a numericalsimulation software package for the producing wellbore, which performsmodeling of the unsteady-state pressure field in the“wellbore-formation” system, flow of non-isothermal fluids in a porousmedium, mixing of the flows in the wellbore, and heat transfer in the“wellbore-formation” system, etc.

Modeling of the process operations carried out under the following timeschedule was performed:

-   -   Circulation of the well during 110 hours. The temperature of        fluids at the formation occurrence depth is assumed to be 40° C.    -   Borehole-return-to-thermal-equilibrium time is 90 hrs.    -   Production for 6 hrs with flow rate Q=60 m³/day.

Geothermal gradient equals 0.02 K/m. The temperature of the undisturbedreservoir at the depth of sensor 1 (274 m) is 65.5° C. and at the depthof sensor 3 (230 m) is 64.6° C. Thermal diffusivity of the reservoir isα=10⁻⁶ m²/s and χ=0.86.

FIG. 1 shows the scheme of a well with three perforated intervals (#1:280-290 m, #2: 260-270 m, #3: 240-250 m) and three temperature sensors:T₁ at the depth of 274 m, T₂ at the depth of 254 m and T₃ at the depthof 230 m.

Two options were considered with different combinations of formationpermeabilities and the following flow rate parameters:

-   Option 1 (FIG. 2 a): Q₁=10 m³/day, Q₂=23.4 m³/day, Q₃=26.6 m³/day;    and-   Option 2 (FIG. 2 b): Q₁=46 m³/day, Q₂32 13 m³/day, Q₃=1 m³/day.

During circulation and the return-to-thermal-equilibrium time, thereservoir/wellbore temperature is the same in both cases underconsideration. At the end of the return-to-thermal-equilibrium time, therate of temperature growth was {dot over (T)}_(s) (200 h)=0.034 K/hr.

FIGS. 3 and 4 show temperatures of the produced fluids (thin curves) andtemperatures of the corresponding sensors (bold curves). The differencebetween T_(in,1) and T₁ remains practically constant after ˜1 hr ofproduction. Time derivatives of fluid temperature and temperature ofsensor #1 are presented in FIGS. 5 and 6. One can see that approximately3 hours after start of production, the difference between dT_(in,1)/dtand {dot over (T)}₁ amounts to about 6-8%, confirming our assumptionmade in the analysis presented above.

The correlation between time derivative T_(in) and specific flow rate q(data for each of the perforated intervals are utilized) is presented inFIG. 9. For flow rate q tending to zero, the linear regression equationgives: {dot over (T)}_(in)(q→0)=0.0374 K/hr. This value is close to therate of temperature recovery {dot over (T)}_(s)(200 h)=0.034 K/hr due tothe conductive heat transfer. This result confirms Formula (14)suggested above for correlation between flow rate and rate oftemperature growth of the produced fluid.

The values of the flow rate can be estimated from the lowermostperforated zone. With duration of production equaling 4 hours, FIGS. 5and 8 give: Option #1—T₁=0.067 K/hr, Option #2—T₁=0.17 K/hr.Substituting these values in Formula (1), we find:

Option #1: Q₁=11 m³/day (the true value is Q₁=10 m³/day);

Option #2: Q₁=46.5 m³/day (the true value is Q₁=46 m³/day).

Flow rate values for other perforated zones are determined by Formulas(20), (23).

Option #1: For the estimated value Q₁=11 m³/day presented above, wefind) y_(a)=1.1. For production duration of 4 hours, FIG. 7 givesf₂₁≈1.45, while Equation (2) gives one positive solution y₂=2.346 andflow rate Q₂=Q₁·y₂=25.8 m³/day.

For the third perforated zone, FIG. 7 gives f₃₂≈1.08 and from Equation(22) we find one positive solution y₃=0.75 and Q₃=(Q₁+Q₂)·y₃=27.6m³/day.

The total flow rate calculated by temperature data amounts toQ_(e)=Q₁+Q₂+Q₃=64.4 m³/day (the true value is 60 m³/day).

Using this value for determining relative flow rates, we find:

${Y_{1} = {\frac{Q_{1}}{Q_{e}} = 0.17}};\mspace{14mu} {Y_{2} = 0.4};\mspace{14mu} {Y_{3} = 0.43}$

The corresponding flow rate values for different zones are:

Q₁=Q·Y₁=10.2 m³/day (the true value is 10 m³/day)

Q₂=Q·Y₂=24 m³/day (the true value is 23.4 m³/day)

Q₁=Q·Y₁=25.8 m³/day (the true value is 26.6 m³/day)

Relative errors (related to the total flow rate) are 0.3%, 1%, and 1.3%.

Option #2: For the above-estimated flow rate value Q₁=46.5 m³/day,y_(a)=0.25. FIG. 8 gives a production duration of 4 hours f₂₁≈0.85. Inthis case, Equation (20) has no solution and as the approximate solutionwe have to take the value of y₂ that corresponds to the minimum value off₂₁ (f_(21 min)≈0.863), which provides for the real solution: y₂=0.413.The corresponding flow rate is Q₂=19.85 m³/day.

For the third perforated zone, Equation 8 gives f₃₂≈0.96 , while fromEquation (22) we find two roots:

-   y₃=0.5, Q₃=(Q₁+Q₂)·y₃=34 m³/day and total flow rate Q_(e)=102    m³/day, and-   y₃=0.062, Q₃=(Q₁+Q₂)·y₃=4.18 m³/day and total flow rate Q_(e)=72    m³/day.

As the approximate solution of the problem, we will take the value ofy₃=0.062, which gives the total flow rate value Q_(e)=72 m³/day that iscloser to the true value.

In the second case the estimate of Q₁ is more reliable than the estimateof Q₂ and Q₃, hence, we fix the value of Q₁ and use the previouslydetermined values of Q₂ and Q₃ for distributing the remaining flow rateQ−Q₁ between these zones:

$Q_{2}^{\prime} = {{\frac{Q_{2}}{Q_{2} + Q_{3}} \cdot \left( {Q - Q_{1}} \right)} = {11.2\mspace{14mu} m^{3}\text{/}{day}}}$and$Q_{3}^{\prime} = {{\frac{Q_{3}}{Q_{2} + Q_{3}} \cdot \left( {Q - Q_{1}} \right)} = {2.3\mspace{14mu} m^{3}\text{/}{day}}}$

The determined flow rate values are as follows:

Q₁=46.5 m³/day (the true value is 46 m³/day)

Q₂=11.2 m³/day (the true value is 13 m³/day)

Q₃=2.3 m³/day (the true value is 1 m³/day)

Relative errors (related to the total flow rate) are 0.8%, 3% and 2.2%.

For solving the inverse problem, this inflow profile (a low inflow rateof the uppermost zone) is complex. Nonetheless, results of solving theinverse problem are consistent with the data utilized in directsimulation.

In general, a reliable inversion of temperature measured amongperforated intervals immediately after perforating can be made with theuse of a specialized numerical model and fitting the transienttemperature data with consideration of absolute values of temperature aswell as time derivatives of temperature.

1. A method for determining profile of fluid inflow from multi-zonereservoirs into a wellbore comprising: measuring a temperature in thewellbore during a wellbore-return-to-thermal-equilibrium time afterdrilling, perforating the wellbore, determining a temperature of thefluids inflowing into the wellbore from each pay zone at an initialstage of production, and determining a specific flow rate for each payzone by a rate of change of the measured temperatures.
 2. The method ofclaim 1, wherein the temperature of the fluids inflowing into thewellbore from the pay zones is determined by a direct measurement oftemperature of the fluids inflowing into the wellbore from each payzone, and a specific flow rate of each pay zone is determined by theformula${Q_{i} = {\frac{4\pi}{\chi} \cdot a \cdot h_{1} \cdot \left( {\frac{{\overset{.}{T}}_{{in},i}}{{\overset{.}{T}}_{s}} - 1} \right)}},$where Q_(i) is a flow rate of the ith pay zone, {dot over (T)}_(s) is arate of temperature recovery in the wellbore before perforation, {dotover (T)}_(in,i) is a rate of temperature variation of the fluidinflowing into the wellbore from the ith pay zone at the initial stageof production, h_(i) is a thickness of the ith pay zone, a is a thermaldiffusivity of the reservoir,${\chi = \frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}},$ ρ_(f)c_(f)is a volumetric heat capacity of the fluid,ρ_(r)c_(r)=φ·ρ_(f)c_(f)+(1−φ)·ρ_(m)c_(m) is a volumetric heat capacityof the rock saturated with the fluid, ρ_(m)c_(m) is a volumetric heatcapacity of a rock matrix, φ is a porosity of the reservoir.
 3. Themethod of claim 1, wherein the wellbore-return-to-thermal-equilibriumtime is 5-10 days.
 4. The method of claim 1, wherein the temperature ofthe fluids inflowing into the wellbore from each pay zone at the initialstage of production is measured within 3-5 hours after start ofproduction.
 5. The method of claim 1, wherein the temperature of thefluids is determined by sensors installed on a tubing string above eachperforated interval, a specific flow rate of a lower pay zone isdetermined by the formula${Q_{1} = {\frac{4\pi}{\chi} \cdot a \cdot h_{1} \cdot \left( {\frac{{\overset{.}{T}}_{1}}{{\overset{.}{T}}_{s}} - 1} \right)}},$where Q₁ is a flow rate of the lower zone, {dot over (T)}_(s) is a rateof temperature recovery in the wellbore before perforation, {dot over(T)}₁ is a rate of temperature change of the fluid inflowing into thewellbore from the pay zone at the initial stage of production asmeasured above the lower perforated interval, h₁ is a thickness of thelower pay zone, a is a thermal diffusivity of the reservoir,${\chi = \frac{c_{f} \cdot \rho_{f}}{\rho_{r} \cdot c_{r}}},$ ρ_(f)c_(f)is a volumetric heat capacity of the fluid,ρ_(r)c_(r)=φ·ρ_(f)c_(f)+(1−φ) ρ_(m)c_(m) is a volumetric heat capacityof the rock saturated by the fluid, ρ_(m)c_(m) is a volumetric heatcapacity of the rock matrix, φ is a porosity of the reservoir, and aspecific flow rate of overlying pay zones is determined by temperaturesmeasured by the sensors installed on the tubing string, using the flowrates determined for the underlying pay zones.
 6. The method of claim 5,wherein the wellbore-return-to-thermal-equilibrium time is 5-10 days. 7.The method of claim 5, wherein the temperature of the fluids inflowinginto the wellbore from each pay zone at the initial stage of productionis measured within 3-5 hours after start of production.